Exact and approximate solutions of the convectiondiffusion equation
Abstract
Even in the case of laminar flows, very few analytic solutions of the convectiondiffusion equation can be found in the literature. In the present investigation, it is shown that, at least in some cases, the convectiondiffusion initialvalue problem can be reduced to the simpler problem of solving a system of nonlinear ordinary differential equations. The employed procedure is based on a utilization of Lie algebra techniques, taking into account a proposal made by Wei and Norman (1963). Lie algebras are introduced with the aid of simple examples, including a model equation for an expanding universe with a timedependent Hubble constant. The considered method is applied to unbounded circular and planar parabolic flow. For the case in which the flow is time dependent, the problem reduces to the solution of a Riccati equation. If the flow is time independent, the solution is given in terms of hyperbolic and trigonometric functions. Bounded flows are also studied.
 Publication:

Quarterly Journal of Mechanics and Applied Mathematics
 Pub Date:
 February 1985
 Bibcode:
 1985QJMAM..38....1P
 Keywords:

 Convection;
 Differential Equations;
 Laminar Flow;
 Lie Groups;
 Molecular Diffusion;
 Riccati Equation;
 Fourier Transformation;
 Harmonic Oscillators;
 Three Dimensional Flow;
 Time Dependence;
 Turbulent Flow;
 Velocity Distribution;
 Fluid Mechanics and Heat Transfer